An \(S\) -flat coherent sheaf \(\mathcal {H}\)  is said to be \(m\) -regular if \(\mathcal {H}_{s}\)  is \(m\) -regular for any geometric point \(s\)  of \(S\) .
There is a relative version of Grothendieck lemma in [[1]}, Lemma 4.13 and Remark 4.14] as follows.
